A balance wheel with crossed strips or beams is a resonator that can be used as a time base in a mechanical watch instead of a sprung balance.
The use of crossed strips or beams has the advantage of increasing the quality factor since there is no longer any friction at the pivot.
However, a balance with crossed strips has two significant drawbacks:                the elastic return torque is non-linear, which makes the system anisochronous, i.e. the frequency of the resonator depends on the amplitude of oscillation;        the centre of mass of the balance is subject to a residual motion which is due to the parasitic motion of the instantaneous axis of rotation. As a result, the resonator frequency depends on the orientation of the watch in the gravitational field; which is known as the position effect.        
In the publication by F. Barrot, T. Hamaguchi, “Un nouveau régulateur mécanique pour une réserve de marche exceptionnelle”, Proceedings of the 2014 Study Day of the Swiss Society of Chronometry, the authors disclosed an oscillator formed of a balance with crossed strips. They explain that “the implementation of a Wittrick type pivot is selected” in order to “make the oscillation frequency independent of the orientation of the balance with respect to gravity”. This particular configuration where the strips intersect at seven eighths of their length was disclosed in the work of W. H. Wittrick, The properties of crossed flexure pivots and the influence of the point at which the strips cross>> The Aeronautical Quarterly II (4), pages 272 to 292 (1951). It has the advantage of minimising the displacements of the virtual axis of rotation and consequently of minimising the position effect. However, with a 90° angle between the two strips, the balance with crossed strips used in these works is highly anisochronous, which is why the authors used compensation via an additional component called the isochronism corrector. Experimental measurements show that such compensation is very difficult to achieve in practice and that it would therefore be very useful to find a geometry for the strips which negates both the position effect and the anisochronism caused by the non-linearity of the elastic return force.
EP Patent Application 2911012A1 in the name of CSEM discloses a rotating timepiece oscillator with a virtual pivot, with a balance that is connected by several flexible strips to a support, particularly in a one-piece embodiment. At least two flexible strips extend in planes perpendicular to the plane of the oscillator, and secant to each other in a straight line defining the geometric axis of oscillation of the oscillator; this axis crosses the two strips at seventh eighths of their respective length.
The configuration with the crossing point at seven eighths of length is already known to be optimum, in order to obtain an own and frictionless rotation about the virtual axis of oscillation, while minimising the displacement of this axis, in accordance with the work of W. H. Wittrick, University of Sydney, February 1951.
Although in this document EP 2 911 012 A1, it is envisaged that the strips emerge perpendicularly to the sides of a regular inner polygon with N sides, with a symmetry of order N about the virtual axis of oscillation, the only specific configuration illustrated is, however, that of an inner square, in which the two planes comprising the strips are perpendicular to each other. According to this document, the number of strips and their arrangement is defined by a compromise between the space allowed for the system, particularly from an aesthetic point of view, and the stability of the system. Apart from the seven eighths rule which is already known, there is no explicit mention in EP Patent Application 2911012 A1 of specific preferred geometric parameters for the best isochronism.